Lie crossed modules and gauge-invariant actions for 2-BF theories

TL;DR

This paper extends BF theory to non-abelian Lie crossed modules, constructing gauge-invariant actions for 2-flat and fake-flat 2-connections, with potential applications in knotted surface invariants.

Contribution

It generalizes BF theory to Lie crossed modules and constructs new gauge-invariant actions for 2-connections, including a three-parameter deformation.

Findings

Constructed gauge-invariant actions for 2-flat and fake-flat 2-connections.

Identified many examples of Lie crossed modules using chain complexes.

Developed a three-parameter deformation relevant for knotted surface invariants.

Abstract

We generalize the BF theory action to the case of a general Lie crossed module $(H \to G)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie algebras of $G$ and $H$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $H$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere.